Optimal single-shot discrimination of optical modes

We first consider the setting for probabilistic discrimination. For uniform priors (our method can be easily generalised to any fixed priors), the optimal guessing probability is given by

$$P_{\text{corr}}^{\text{opt}}=\max_{\{M_{j}\},\{c_{n}\}}\frac{1}{N}\sum_{j=1}^{N}\bra{\psi_{j}}M_{j}\ket{\psi_{j}}$$

(4)

where $M_{j}$ is the POVM element associated with the outcome $j$ and $c_{n}$ is the coefficient of the state when written in the photon number basis. In other words, we have to optimize both the state and the measurements that maximize the guessing probability, subject to the energy constraint. Let us now cast this optimization into a SDP by adapting the techniques of Wang et al. (2019). Before getting to it, we notice that Wang et al. (2019); Navascués et al. (2007, 2008) considers a hierarchy of semidefinite relaxations, which in general only yield upper bounds on the guessing probability. However, since we only consider a single receiver with no classical inputs, going into the second level of the hierarchy will satisfy the rank loop condition Navascués et al. (2008) and hence the first level of the hierarchy is actually tight.

Here comes the construction. Consider the set $\mathcal{O}=\{\mathbb{1},M_{1},...,M_{N}\}$. As discussed in Wang et al. (2019); Navascués et al. (2007, 2008), the $\{M_{i}\}$ can be taken as projective measurements without any loss of generality. Denoting $O_{i}$ the elements of $\mathcal{O}$, one can define a set of vectors $\mathcal{S}=\{O_{i}\ket{\psi_{j}}:O_{i}\in\mathcal{O},\ket{\psi_{j}}\in\mathcal{R}\}$. Since all Gram matrices are positive semidefinite (PSD), so is the $N(N+1)\times N(N+1)$ Gram matrix $G$ associated to the set $\mathcal{S}$. Hence we have

$$\begin{split}P_{\text{corr}}^{\textrm{opt}}=&\max_{G,\{p_{n}\}}\frac{1}{N}\sum_{j=1}^{N}\bra{\psi_{j}}M_{j}\ket{\psi_{j}}\\ \text{s.t.}\ &p_{n}\geq 0\qquad\forall n\\ &\sum_{n}p_{n}=1\\ &\sum_{n}p_{n}n=\bar{n}\\ &G\succeq 0\\ &\innerproduct{\psi_{i}}{\psi_{j}}=\sum_{n}p_{n}k_{ij}^{n}\qquad\forall i,j\,,\end{split}$$

(5)

where the last relation is Eq. (3) and determines the entries of $G$ associated with $O_{0}=\mathbb{1}$.

However, this is an optimization problem with infinitely many variables $p_{n}$ and hence is computationally intractable. We then relax it by truncating the number of photons to $n_{\text{max}}$ (i.e. we’ll have $n_{\text{max}}+1$ variables $p_{n}$). We do it in such a way as to obtain an upper bound $P_{\text{corr}}^{\textrm{SDP}}\geq P_{\text{corr}}^{\text{opt}}$ on the mode discrimination probabilities: that is, we derive some necessary (but not sufficient) conditions on the photon number distribution, and as a result the feasible region may be larger than the one allowed by quantum theory. Clearly, the relaxation can be made arbitrarily tight by increasing the photon number cut-off; and we expect $P_{\text{corr}}^{\text{opt}}\approx P_{\text{corr}}^{\textrm{SDP}}$ when $\bar{n}\ll n_{\text{max}}$.

We then define the truncated state

$$\ket{\tilde{\psi}_{j}}=\sum_{n=0}^{n_{\text{max}}}c_{n}\frac{1}{\sqrt{n!}}{a_{j}^{\dagger}}^{n}\ket{0}$$

(6)

that is sub-normalised:

$$\sum_{n=0}^{n_{\text{max}}}p_{n}\leq 1\,.$$

(7)

The inner-product of the truncated states is

$$\innerproduct{\tilde{\psi_{i}}}{\tilde{\psi_{j}}}=\sum_{n=0}^{n_{\text{max}}}p_{n}k_{ij}^{n}$$

(8)

and its difference with the inner product of the full states can be bounded as

$$\begin{split}\absolutevalue{\innerproduct{\tilde{\psi_{i}}}{\tilde{\psi_{j}}}-\innerproduct{\psi_{i}}{\psi_{j}}}&=\absolutevalue{\sum_{n>n_{\text{max}}}p_{n}k_{ij}^{n}}\\ &\leq\sum_{n>n_{\text{max}}}p_{n}\absolutevalue{k_{ij}}^{n}\\ &\leq\left(1-\sum_{n=0}^{n_{\text{max}}}p_{n}\right)\absolutevalue{k_{ij}}^{n_{\text{max}}+1}\equiv\varepsilon_{ij}\end{split}$$

(9)

where the first inequality is a consequence of triangle inequality and the second inequality is due to the fact that $\absolutevalue{k_{ij}}\leq 1$. The constraint on the mean photon number can be relaxed to

$$\sum_{n=0}^{n_{\text{max}}}p_{n}n+\left(n_{\text{max}}+1\right)\left(1-\sum_{n=0}^{n_{\text{max}}}p_{n}\right)\leq\bar{n}\,.$$

(10)

All in all, we have $P_{\text{corr}}^{\text{opt}}\leq P_{\text{corr}}^{\textrm{SDP}}$ with

$$\begin{split}P_{\text{corr}}^{\textrm{SDP}}=&\max_{G,\{p_{n}\}}\frac{1}{N}\sum_{j=1}^{N}\bra{\psi_{j}}M_{j}\ket{\psi_{j}}\\ \text{s.t.}\ &G\succeq 0\\ &p_{n}\geq 0\qquad\forall n\leq n_{\text{max}}\\ &\sum_{n\leq n_{\text{max}}}p_{n}\leq 1\\ &\sum_{n\leq n_{\text{max}}}p_{n}(n_{\text{max}}+1-n)\geq n_{\text{max}}+1-\bar{n}\\ &\absolutevalue{\innerproduct{\psi_{i}}{\psi_{j}}-\innerproduct{\tilde{\psi_{i}}}{\tilde{\psi_{j}}}}\leq\varepsilon_{ij}\qquad\forall i,j\end{split}$$

(11)

where the last constraint uses the expressions (8) and (9). That constraint is linear in the $p_{n}$ when $k_{ij}$ is real; when $k_{ij}$ is complex, we can re-write it as

$$\begin{pmatrix}\varepsilon_{ij}&\innerproduct{\psi_{i}}{\psi_{j}}-\innerproduct{\tilde{\psi_{i}}}{\tilde{\psi_{j}}}\\ \left(\innerproduct{\psi_{i}}{\psi_{j}}-\innerproduct{\tilde{\psi_{i}}}{\tilde{\psi_{j}}}\right)^{*}&\varepsilon_{ij}\end{pmatrix}\succeq 0$$

(12)

which is a matrix inequality linear in the $p_{n}$, hence a valid SDP constraint.

The same technique can be adapted to find the maximum success probability for unambiguous mode discrimination. For unambiguous discrimination, one must allow for the inconclusive outcome, which we associate to the POVM element $M_{\varnothing}$. The Gram matrix $G$ will correspondingly increase in size to $N(N+2)\times N(N+2)$. Then we must add the constraint that the probability of error is zero

$$\sum_{j=1}^{N}\sum_{i\neq j,i\neq\varnothing}\bra{\psi_{j}}M_{i}\ket{\psi_{j}}=0$$

(13)

In the study of unambiguous discrimination, the aim is to minimize the probability of the inconclusive outcome or, equivalently, to maximize the success probability. Assuming uniform priors, the success probability is given by

$$P_{\text{UD}}=1-\frac{1}{N}\sum_{j=}^{N}\bra{\psi_{j}}M_{\varnothing}\ket{\psi_{j}}$$

(14)

Note that both the success probability and the error probability are linear functions of the Gram matrix $G$. Hence, putting everything together, we have $P_{\text{UD}}^{\text{opt}}\leq P_{\text{UD}}^{\textrm{SDP}}$ with

$$\begin{split}P_{\text{UD}}^{\textrm{SDP}}=&\max_{G,\{p_{n}\}}1-\frac{1}{N}\sum_{j=1}^{N}\bra{\psi_{j}}M_{\varnothing}\ket{\psi_{j}}\\ \text{s.t.}\ &p_{n}\geq 0\qquad\forall n\leq n_{\text{max}}\\ &\sum_{n\leq n_{\text{max}}}p_{n}\leq 1\\ &\sum_{n\leq n_{\text{max}}}p_{n}(n_{\text{max}}+1-n)\geq n_{\text{max}}+1-\bar{n}\\ &G\succeq 0\\ &\absolutevalue{\innerproduct{\psi_{i}}{\psi_{j}}-\innerproduct{\tilde{\psi_{i}}}{\tilde{\psi_{j}}}}\leq\varepsilon_{ij}\qquad\forall i,j\\ &\sum_{j=1}^{N}\sum_{i\neq j,i\neq\varnothing}\bra{\psi_{j}}M_{i}\ket{\psi_{j}}=0\end{split}$$

(15)

Recall that unambiguous state discrimination is possible if and only if the states to be discriminated are linearly independent. But unambiguous mode discrimination is possible even for linearly dependent modes, as the linear independence of the states is provided by the multiphoton components. In fact, the families studied in subsections IV.2, IV.3 and IV.4 will be linearly dependent.

For probabilistic mode discrimination, the guessing probability is given by

	$\displaystyle P_{\text{corr}}^{\text{opt}}$	$\displaystyle=\max_{\{p_{n}\},\{M_{j}\}}\frac{1}{N}\sum_{j=1}^{N}\Tr(\rho_{j}M_{j})$
		$\displaystyle\equiv\max_{\{p_{n}\}}\sum_{n=0}^{\infty}p_{n}P_{\text{corr}}^{(n)}$		(18)

where

$$P_{\text{corr}}^{(n)}=\max_{\{M_{j}\}}\frac{1}{N}\sum_{j=1}^{N}\bra{n_{j}}M_{j}\ket{n_{j}}$$

(19)

is the optimal guessing probability for $n$ photons. Therefore, as expected, we can split the optimization into two steps. In the first step, we solve (19) for each value of $n$. This can be done using the SDP technique of Ref. Wang et al. (2019). Similar to what we have done in the channel-discrimination scenario, consider the set $\mathcal{R}_{n}=\{\ket{n_{1}},\ket{n_{2}},...,\ket{n_{N}}\}$ which are Fock state $n$ from the set of modes $\mathcal{M}$. We define the set of vectors $\mathcal{S}_{n}=\{O_{i}\ket{n_{j}}:O_{i}\in\mathcal{O},\ket{n_{j}}\in\mathcal{R}_{n}\}$. Now, denote the Gram matrix associated to $\mathcal{S}_{n}$ by $G^{(n)}$. The SDP to bound $P_{\text{corr}}^{(n)}$ is

$$\begin{split}P_{\text{corr}}^{(n)}=&\max_{G^{(n)}}\frac{1}{N}\sum_{j=1}^{N}\bra{n_{j}}M_{j}\ket{n_{j}}\\ &\text{s.t. }\ G^{(n)}\succeq 0\\ &\qquad\innerproduct{n_{i}}{n_{j}}=k_{ij}^{n}\end{split}$$

(20)

which we have to solve for each photon number $n$. In the second step, having $P_{\text{corr}}^{(n)}$ for each photon number $n$, we just need to enforce the energy constraint. This remaining step is linear programming (LP):

$$\begin{split}P_{\text{corr}}^{\text{opt}}=&\max_{\{p_{n}\}}\sum_{n}p_{n}P_{\text{corr}}^{(n)}\\ &\text{s.t. }\ p_{n}\geq 0\qquad\forall n,\\ &\qquad\sum_{n}p_{n}=1,\\ &\qquad\sum_{n}p_{n}n=\bar{n}\end{split}$$

(21)

Like in the channel-discrimination scenario, we have infinitely many variables $p_{n}$, so we need a cutoff that relaxes the original LP. With the same arguments as above, the resulting relaxation $P_{\text{corr}}^{\text{LP}}\geq P_{\text{corr}}^{\text{opt}}$ is given by

$$\begin{split}P_{\text{corr}}^{\text{LP}}=&\max_{\{p_{n}\}}\sum_{n=0}^{n_{\text{max}}}p_{n}P_{\text{corr}}^{(n)}+\left(1-\sum_{n=0}^{n_{\text{max}}}p_{n}\right)\\ &\text{s.t. }\ p_{n}\geq 0\qquad\forall n\leq n_{\text{max}},\\ &\qquad\sum_{n=0}^{n_{\text{max}}}p_{n}\leq 1,\\ &\qquad\sum_{n=0}^{n_{\text{max}}}p_{n}n+\left(1-\sum_{n=0}^{n_{\text{max}}}p_{n}\right)(n_{\text{max}}+1)\leq\bar{n}\end{split}$$

(22)

Notice that this relaxation is equivalent to assuming that the modes are perfectly distinguishable for $n>n_{\text{max}}$. This is a good approximation when we pick sufficiently high photon number cutoff $n_{\text{max}}$.

Also for unambiguous state discrimination the optimal success probability is of the form

$$P_{\text{UD}}^{\text{opt}}=\max_{\{p_{n}\}}\sum_{n}p_{n}P_{\text{UD}}^{(n)}$$

(23)

and can be bounded in two steps. In the first step, $P_{\text{UD}}^{(n)}$ is computed from the SDP

$$\begin{split}P_{\text{UD}}^{(n)}=&\max_{G^{(n)}}1-\frac{1}{N}\sum_{j=1}^{N}\bra{n_{j}}M_{\varnothing}\ket{n_{j}}\\ &\text{s.t. }\ G^{(n)}\succeq 0\\ &\qquad\innerproduct{n_{i}}{n_{j}}=k_{ij}^{n}\\ &\qquad\bra{n_{j}}M_{i}\ket{n_{j}}=0,\quad\forall i\neq j,i\neq\varnothing\end{split}$$

(24)

where the last constraint captures the unambiguous discrimination condition

$$\sum_{j=1}^{N}\sum_{i\neq j,i\neq\varnothing}\Tr(\rho_{j}M_{i})=0\,,$$

(25)

which is indeed satisfied if and only if $\bra{n_{j}}M_{i}\ket{n_{j}}=0$ for all $n$ whenever $i\neq\varnothing$, $j\neq i$. In the second step, the energy constraint is enforced in a LP, which we write down directly for the relaxation $P_{\text{UD}}^{\text{LP}}\geq P_{\text{UD}}^{\text{opt}}$ with a photon-number cutoff:

$$\begin{split}P_{\text{UD}}^{\text{LP}}=&\max_{\{p_{n}\}}\sum_{n=0}^{n_{\text{max}}}p_{n}P_{\text{UD}}^{(n)}+\left(1-\sum_{n=0}^{n_{\text{max}}}p_{n}\right)\\ &\text{s.t. }\ p_{n}\geq 0\qquad\forall n\leq n_{\text{max}},\\ &\qquad\sum_{n=0}^{n_{\text{max}}}p_{n}\leq 1,\\ &\qquad\sum_{n=0}^{n_{\text{max}}}p_{n}n+\left(1-\sum_{n=0}^{n_{\text{max}}}p_{n}\right)(n_{\text{max}}+1)\leq\bar{n}\,.\end{split}$$

(26)

We have just seen that the final step of the optimization for the source-discrimination scenario is a LP of the form

$$\begin{split}P^{\text{opt}}=&\max_{\{p_{n}\}}\sum_{n}p_{n}\mathsf{a}_{n}\\ &\text{s.t. }\ p_{n}\geq 0\qquad\forall n,\\ &\qquad\sum_{n}p_{n}=1,\\ &\qquad\sum_{n}p_{n}n=\bar{n}\end{split}$$

(27)

where $\mathsf{a}_{n}=P_{\text{corr}}^{(n)}$ for probabilistic discrimination and $\mathsf{a}_{n}=P_{\text{UD}}^{(n)}$ for unambiguous discrimination. For a given $\mathsf{a}_{n}$, this LP can be solved analytically. We are going to show how this can be done, and highlight a condition on $\mathsf{a}_{n}$ under which the solution can be easily spelled out.

The LP (27) is written in the so-called primal form. One could also consider its dual form given by Boyd and Vandenberghe (2004)

$$\begin{split}d^{*}=\min_{x,y}\ &x+\bar{n}y\\ \text{s.t.}\,\,&y\geq-\frac{1}{n}\left(x-\mathsf{a}_{n}\right)\quad\forall n\\ \end{split}$$

(28)

Due to the strong duality of LP, we know that $d^{*}=P^{\text{opt}}$ and hence it is sufficient to solve the dual problem.

Whereas the primal problem is an optimization over infinitely many variables with a few equality constraints, the dual problem is an optimization over two variables with infinitely many constraints, which define the feasible region. The way to solve the dual problem is easily understood geometrically (Fig. 2). One traslates the lines $L_{\text{obj}}(d)=\{(x,y)\;|\,x+\bar{n}y=d\}$ with fixed gradient $-1/\bar{n}$, till finding the lowest one that has at least one point in the feasible region. But the boundary of the feasible region is given by segments of straight lines $L_{n}$ of gradient $-1/n$. So the limiting line $L_{\text{obj}}(d^{*})$ may touch the boundary of the feasible set either in a single point (the intersection of two $L_{n}$, as illustrated in the figure) or in a whole segment. The latter can only happen if $\bar{n}=n$, but this is not the only condition: it is further necessary that $L_{n}$ contributes to the boundary of the feasible region. This may not always be the case, as illustrated in Fig. 2. By this recipe, one can always find the solution, given the $\mathsf{a}_{n}$.

It is worth describing in detail the case where all the constraints in (28), i.e. all the $L_{n}$, contribute to the boundary of the feasible region in a non-trivial way. Given that the gradient of $L_{n}$ is $-1/n$, a necessary and sufficient condition for the boundary to be as described is that $x_{n-1,n}<x_{n,n+1}$ where $(x_{n,m},y_{n,m})$ are the coordinates of the intersection of $L_{n}$ with $L_{m}$. From $y_{n-1,n}=-\frac{1}{n-1}(x_{n-1,n}-\mathsf{a}_{n-1})=-\frac{1}{n}(x_{n-1,n}-\mathsf{a}_{n})$ one immediately finds $x_{n-1,n}=n\mathsf{a}_{n-1}-(n-1)\mathsf{a}_{n}$. Thus, $x_{n-1,n}<x_{n,n+1}$ will be the case if and only if

$$\mathsf{a}_{n-1}-2\mathsf{a}_{n}+\mathsf{a}_{n+1}<0\,.$$

(29)

If this condition is satisfied, then the optimal discrimination takes up a very clear form. Indeed, for a boundary as described:

• (i)

  If $\bar{n}\not\in\mathbb{N}$, the intersection that defines $d^{*}$ will be with a single point, namely the intersection of $L_{\lfloor\bar{n}\rfloor}$ and $L_{\lceil\bar{n}\rceil}$. The optimal state is then a mixture of two Fock states with these numbers, and the suitable weights.

• (ii)

  If $\bar{n}=n\in\mathbb{N}$, the intersection that defines $d^{*}$ will be the whole segment of gradient $-1/n$, and the optimal state with be the Fock state $\rho=\ket{n}\bra{n}$.

In other words, the solution of the LP will be $P^{\textrm{opt}}=p_{\lfloor\bar{n}\rfloor}\mathsf{a}_{\lfloor\bar{n}\rfloor}+p_{\lceil\bar{n}\rceil}\mathsf{a}_{\lceil\bar{n}\rceil}$, with

$$p_{n}=\begin{cases}1+\lfloor\bar{n}\rfloor-\bar{n}&\text{if $n=\lfloor\bar{n}\rfloor$,}\\ \bar{n}-\lfloor\bar{n}\rfloor&\text{if $n=\lfloor\bar{n}\rfloor+1$,}\\ 0&\text{otherwise}\,.\end{cases}$$

(30)

Thus, in many cases we can expect the optimal state for discrimination to consist of the Fock state $\ket{\bar{n}}$ if $\bar{n}\in\mathbb{N}$, and of the suitable mixture of the Fock states $\ket{\lfloor\bar{n}\rfloor}$ and $\ket{\lceil\bar{n}\rceil}$ if $\bar{n}\not\in\mathbb{N}$. However, condition (29) does not always hold: in subsection IV.3 we shall see an example where it is not met for $n=1$, and indeed the Fock state $\ket{1}$ will not be optimal for $\bar{n}=1$.

The discrimination between two modes is determined by a single parameter

$$[a_{1},a_{2}^{\dagger}]=k\mathbb{1}\;,\quad k\in\mathbb{C}\,,\;|k|\leq 1\,.$$

(31)

When $k=1$, the two modes are identical and therefore indistinguishable. When $k=0$, the two modes are orthogonal and can be perfectly distinguished when $\bar{n}\geq 1$.

Besides using our numerical tools, we are going to derive analytical solutions, exploiting the fact that the discrimination of two equally probable pure states has been solved long ago for both probabilistic Helstrom (1969) and unambiguous discrimination Ivanovic (1987); Dieks (1988); Peres (1988):

$$\begin{split}P_{\text{corr}}^{\text{opt}}&=\frac{1}{2}(1+\sqrt{1-\absolutevalue{\langle\psi_{1}|\psi_{2}\rangle}^{2}})\\ P_{\text{UD}}^{\text{opt}}&=1-\absolutevalue{\langle\psi_{1}|\psi_{2}\rangle}\,.\end{split}$$

(32)

With this, single-shot discrimination of two modes in the source-discrimination scenario can be fully solved analytically. Indeed, there is no need to solve the SDPs (20) and (24) since we know from (32) that $P_{\text{corr}}^{(n)}=\frac{1}{2}(1+\sqrt{1-\absolutevalue{k}^{2n}})$ and $P_{\text{UD}}^{(n)}=1-\absolutevalue{k}^{n}$ (notice that everything depends on $\absolutevalue{k}$.). Moving to the LP, it is immediate to verify that both expressions satisfy condition (29). So we can import from subsection III.3 that the solution is

$$P_{\textrm{corr/UD}}^{\textrm{opt}}=p_{\lfloor\bar{n}\rfloor}P_{\textrm{corr/UD}}^{(\lfloor\bar{n}\rfloor)}+p_{\lceil\bar{n}\rceil}P_{\textrm{corr/UD}}^{(\lceil\bar{n}\rceil)}$$

(33)

with the $p_{n}$ given in (30).

In the channel-discrimination scenario, we know from (32) that $P_{\text{corr}}^{\text{opt}}=\frac{1}{2}(1+\sqrt{1-\chi^{2}})$ and $P_{\text{UD}}^{\text{opt}}=1-\chi$ with

$$\begin{split}\chi=&\min_{\{p_{n}\}}\left|\sum_{n=0}^{\infty}p_{n}k^{n}\right|\\ &\text{s.t. }\ p_{n}\geq 0\qquad\forall n,\\ &\qquad\sum_{n}p_{n}=1,\\ &\qquad\sum_{n}p_{n}n=\bar{n}\end{split}$$

(34)

where we used the expression (3) of the scalar product. Instead of the SDPs, we could try and solve (34).

For $k\geq 0$, it is a LP of the form (28) with $\mathsf{a}_{n}\equiv-k^{n}$ (notice that here we are minimizing, whence the sign). Condition (29) reads $-k^{n-1}(1+k^{2})<0$ and is therefore satisfied: so we know that the solution is

$\displaystyle\chi=p_{\lfloor\bar{n}\rfloor}k^{\lfloor\bar{n}\rfloor}+p_{\lceil\bar{n}\rceil}k^{\lceil\bar{n}\rceil}$

$\displaystyle[k\geq 0]$

(35)

with the $p_{n}$ given in (30). The corresponding optimal state is $\sqrt{p_{\lfloor\bar{n}\rfloor}}\ket{\lfloor\bar{n}\rfloor}+e^{i\varphi}\sqrt{p_{\lceil\bar{n}\rceil}}\ket{\lceil\bar{n}\rceil}$ for any $\varphi$. This value of $\chi$ shows that $P_{\text{corr}}^{\textrm{opt}}$ is larger than in the source-discrimination scenario (33), since $\sqrt{1-\chi^{2}}\geq p_{\lfloor\bar{n}\rfloor}\sqrt{1-k^{2\lfloor\bar{n}\rfloor}}+p_{\lceil\bar{n}\rceil}\sqrt{1-k^{2\lceil\bar{n}\rceil}}$; while the value of $P_{\text{UD}}^{\textrm{opt}}$ is identical in the two scenarios.

For $k<0$, the optimization (34) is also LP; but whether the absolute value adds a minus sign or not (i.e. whether $\mathsf{a}_{n}=+k^{n}$ or $-k^{n}$) is not known a priori. One would therefore have to solve the two LPs, then compare the solutions. In either case, condition (29) would be satisfied only for alternate $n$, and so the solution is not expected to involve only $\lfloor\bar{n}\rfloor$ and $\lceil\bar{n}\rceil$. As for $k\in\mathbb{C}\setminus\mathbb{R}$, the optimization (34) is quadratic, and there is no guarantee that an analytical solution can be found.

We thus turn to our SDPs (11) and (15). The results are shown in Fig. 3 for probabilistic discrimination, and in Fig. 4 for unambiguous discrimination. As expected, the modes are harder to distinguish when $k\approx 1$. More remarkable is the fact that distinguishability improves significantly in the region of negative phases. For instance, while it is known and rather obvious that perfect discrimination for $k=0$ becomes possible for $\bar{n}\geq 1$, we see that when $k=-1$ the modes can already be perfectly distinguished for $\bar{n}=0.5$ (more in Section IV.2).

Next we consider the problem of phase discrimination. Since the phase of an optical mode is not defined unless a reference beam is provided, this case study is restricted to the channel-discrimination scenario. The receiver’s task is to guess one of a set of unitary channels $U_{j}=e^{i\varphi_{j}\hat{n}}$ where $\hat{n}$ is the number operator in the signal mode. In other words, the receiver can use pure states to distinguish modes of the form

$$a_{j}=e^{i\varphi_{j}}a$$

(36)

where $a$ is the initial signal mode prior to the phase-shift.

Our formalism can be applied to any set of phases to be discriminated. For the numerical case study, we choose the symmetric set $\{\varphi_{j}=2\pi j/N|j=0,...,N-1\}$, whose probabilistic discrimination has been studied in the context of quantum sensing  Nair et al. (2012). The commutation relations are then given by

$$[a_{j},a_{k}^{\dagger}]=e^{i2\pi(j-k)/N}\mathbb{1}\,.$$

(37)

For probabilistic discrimination (see Fig. 5(a)), our SDP (11) recovers the same bound as Theorem 3 of Ref. Nair et al. (2012). The study of unambiguous discrimination is novel: the result is plotted in Fig. 5(b). Also notice that, for $N>2$, the modes in this family are linearly dependent; but, as discussed, unambiguous discrimination is possible because one can create linearly independent states.

The $N=2$ case corresponds to $k=-1$ in Section IV.1. As we notice there and see again here, perfect discrimination becomes possible at $\bar{n}=0.5$. This is because the pure states with $c_{0}=c_{1}=\frac{1}{\sqrt{2}}$ are orthogonal. Indeed, $\ket{\psi_{a}}=\frac{1}{\sqrt{2}}(\ket{0}+a^{\dagger}\ket{0})=\frac{1}{\sqrt{2}}(\ket{0}+\ket{1}_{a})$, while $\ket{\psi_{-a}}=\frac{1}{\sqrt{2}}(\ket{0}+(-a^{\dagger})\ket{0})=\frac{1}{\sqrt{2}}(\ket{0}-\ket{1}_{a})$. Since distinguishability cannot decrease when increasing $\bar{n}$, one expects to find two orthogonal states for any $\bar{n}\geq 0.5$. Indeed, these are $\sqrt{\frac{1-\delta}{4}}\ket{m-1}_{a}\pm\frac{1}{\sqrt{2}}\ket{m}_{a}+\sqrt{\frac{1+\delta}{4}}\ket{m+1}_{a}$ with $\bar{n}=m+\delta/2$ and $-1\leq\delta<1$. In general these are superpositions of three Fock states, reducing to two only when $\bar{n}=m-\frac{1}{2}$ ($\delta=0$). Thus, as anticipated, the optimal state does not obey (30).

As our next example, we consider a family made of two sets of orthogonal modes. The first set $\mathcal{C}_{d}=\{a_{0},a_{1},...,a_{d-1}\}$ are called computational basis modes. The second set $\mathcal{F}_{d}=\{b_{0},b_{1},...,b_{d-1}\}$ are called Fourier transform basis modes and are defined as

$$b_{k}=\frac{1}{\sqrt{d}}\sum_{j=0}^{d-1}(\omega_{d})^{jk}a_{j}$$

(38)

where $\omega_{d}=e^{2\pi i/d}$ is the $d$-th root of unity. This family of modes have appeared in quantum cryptography: the classical information is encoded into these modes in the famous BB84 protocol Bennett and Brassard (1984) for $d=2$, and the case $d>2$ defines one of its possible generalizations to higher alphabets Sheridan and Scarani (2010). However, in those QKD protocols the classical information is encoded in the mode’s index: one wants to determine $j$, whether from $a_{j}$ or $b_{j}$. By contrast, here we stay with the task of mode discrimination, and study both probabilistic and unambiguous discrimination from the set $\mathcal{M}_{d}=\mathcal{C}_{d}\cup\mathcal{F}_{d}$.

For given $d$, the commutation relations are

$$\begin{split}[a_{j},a_{k}^{\dagger}]&=\delta_{jk}\mathbb{1}\\ [b_{j},b_{k}^{\dagger}]&=\delta_{jk}\mathbb{1}\\ [a_{j},b_{k}^{\dagger}]&=\frac{1}{\sqrt{d}}\left(\omega_{d}^{*}\right)^{jk}\mathbb{1}\\ \end{split}$$

(39)

We set $n_{\text{max}}=50$ and solve the SDP and/or LP for $\bar{n}$ varying from $10^{-3}$ to 10, and for $d=2,3,4,5$. The results for probabilistic discrimination are shown in Fig. 6(a); those for unambiguous discrimination in 6(b); both figures contain information about the channel-discrimination scenario (dashed) and the source-discrimination scenario (solid).

Channel discrimination is more powerful than source discrimination: while more marked for probabilistic discrimination, the difference is also present in unambiguous discrimination, contrary to what was the case for two modes (Section IV.1). Another feature present in both figures, again more marked for probabilistic discrimination, is a crossover of behavior as a function of $\bar{n}$: for $\bar{n}\lesssim 1$, the discrimination is better for smaller $d$; whereas for $\bar{n}\gtrsim 1$, the discrimination is better for higher $d$. This can be understood qualitatively. In the limit $\bar{n}\rightarrow 0$, guessing the mode is hardly more than a random pick from a uniform distribution, so the guessing probability is close to $\frac{1}{2d}$ and decreases with $d$. In the limit of large $n$, $|\innerproduct{n_{a_{j}}}{n_{b_{k}}}|=\sqrt{1/d^{n}}$ decreases with $d$ and therefore the distinguishability increases.

In the source-discrimination scenario, we find that condition (29) is generally not satisfied for $\bar{n}=1$, either for probabilistic or unambiguous discrimination.

For unambiguous discrimination, the reason is clear: the single-photon states are linearly dependent and hence unambiguous discrimination is not possible. In order for unambiguous discrimination to be possible, the state must contain some multi-photon component.

For probabilistic discrimination, we find that the condition (29) is violated for $d=3,4,5$. To see that, we compare $P_{\text{corr}}^{\text{LP}}$ with that of Fock states in Fig. 7. For $d=2,3,4,5$, the single-photon bound is $P_{\text{corr}}^{(1)}=0.5$. This corresponds to a simple strategy: betting on one of the bases and measure in it.

Lastly, we consider a family of modes inspired by the differential-phase-shift (DPS) QKD protocol Inoue et al. (2002). In the protocol, the information is encoded in the relative phase between subsequent temporal modes. Abstractly, to any $\ell$-bit strings $\mathbf{x}$, a mode is associated according to

$$b_{\mathbf{x}}=\frac{1}{\sqrt{\ell+1}}\left(a_{0}+\sum_{i=1}^{\ell}e^{i\varphi_{i}^{(\mathbf{x})}}a_{i}\right),$$

(40)

where the $\{a_{i}\}_{i=0,...,\ell}$ are $(\ell+1)$ orthogonal modes (temporal ones in the original setting) and where

$$\varphi_{i}^{(\mathbf{x})}-\varphi_{i-1}^{(\mathbf{x})}=x_{i}\pi$$

(41)

with $x_{i}\in\{0,1\}$ the $i$-th bit of the string $\mathbf{x}$ (by convention, we set the phase of the reference mode $\varphi_{0}^{(\mathbf{x})}=0$ for all $\mathbf{x}$). For a given $\ell$, there are $2^{\ell}$ different modes, only linearly-many of which are orthogonal among the $2^{\ell}$ ones. The commutation relation for the modes associated to string $\mathbf{x}$ and $\mathbf{y}$ can be computed recursively using the relation (41) and is given by

$$\left[b_{\mathbf{x}},b^{\dagger}_{\mathbf{y}}\right]=\frac{1}{\ell+1}\left(1+\sum_{i=1}^{\ell}e^{i(\varphi_{i}^{(\mathbf{x})}-\varphi_{i}^{(\mathbf{y})})}\right)\mathbb{1}\,.$$

(42)

Since the SDP scales badly with $\ell$, in this paper we present the results only for $\ell=1,2,3$. When $\ell=1$, the two modes to be distinguished are orthogonal, and so this is a special case of what we studied in Section IV.1. For $\ell=2$, the four modes are all non-orthogonal. The eight modes for $\ell=3$ form two sets of four orthogonal modes.

The results of our numerical method are shown in Fig. 8(a) for probabilistic discrimination, and in Fig. 8(b) for unambiguous discrimination. Since the family given by $\ell$ is constructed from $\ell+1$ orthogonal modes (“pulses”), we found it more appropriate to compare the families for a given value of the energy per pulse $\mu=\bar{n}/(\ell+1)$, rather than of the total energy $\bar{n}$. Even with this scaling, it proves more difficult to distinguish within a set with higher $\ell$, since the receiver has to discriminate more modes.

Comparing the mixed state encoding to the pure state encoding, we find that the pure state encoding is more distinguishable only in the probabilistic discrimination setting. Remarkably, we find that the mixed state bounds coincide with that of the pure state encoding for the unambiguous discrimination setting.